Abstract

We study a theoretical model describing a laser with a modulated parameter, concentrating on the appearance of extreme events, also called optical rogue pulses. It is shown that two conditions are required for the appearance of such events in this type of nonlinear system: the existence of generalized multi-stability and the collisions of chaotic attractors with unstable orbits in external crisis, expanding the attractor to visit new regions in phase space.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. L. Borges, “Siete Noches” - La Divina comedia; Nueve Ensayos Dantescos, Ed. Fondo de Cultura Economica, Buenos Aires (1980).
  2. P. S. Laplace, A Philosophical Essay on Probability, translated (John Wiley, 1902).
  3. S. Aberg and G. Lindgren, “Height distribution of stochastic Lagrange ocean waves,” Prob. Eng. Mech. 23, 359–363 (2008).
    [CrossRef]
  4. A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
    [CrossRef]
  5. U. F. de Pinho, P. C. Liu, and C. E. P. Ribeiro, “Freak waves at Campos Basin, Brazil,” Geofizika 21, 53 (2004)
  6. S. K. El-Labany, W. M. Moslem, N. A. El-Bedwehy, R. Sabry, and H. N. A. El-Razek, “Rogue waves in Titan’s atmosphere,” Astrophys. Space Sci. 338, 3–8 (2012).
    [CrossRef]
  7. H. Kawamura, T. Hatano, N. Kato, S. Biswas, and B. K. Chakrabartik, “Statistical physics of fracture, friction, and earthquakes,” Rev. Mod. Phys. 84, 839–884 (2012).
    [CrossRef]
  8. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
    [CrossRef] [PubMed]
  9. K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
    [CrossRef]
  10. C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
    [CrossRef] [PubMed]
  11. C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 87, 2339108 (2013).
  12. M. Kovalsky, A. Hilo, and J. R. Tredicce, “Extreme events in the Ti:sapphire laser,” Opt. Lett. 36, 4449–4451 (2011)
    [CrossRef] [PubMed]
  13. W. H. Munk, “Proposed uniform procedure for observing waves and interpreting instruments records,” Wave Project at the Scripps Institute of Oceanography, Ca. USA (1944).
  14. P. Kjeldsen, “A sudden disaster in extreme waves,” Rogue Waves 2000, IFREMER (1935).
  15. J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
    [CrossRef]
  16. C. Bonazzola, A. Hnilo, M. Kovalsky, and J. R. Tredicce, “Optical rogue waves in an all-solid state laser with saturable absorber: importance of spatial effects,” J. Opt. 15, 064004 (2013).
    [CrossRef]
  17. P. Gaspard and X.-J. Wang, “Sporadicity: Between periodic and chaotic dynamical behaviors,” Proc. Natl. Acad. Sci. USA 85, 4591–4595 (1988).
    [CrossRef] [PubMed]
  18. V. Balakrishnan, C. Nicolis, and G. Nicolis, “Extreme value distributions in chaotic dynamics,” J. Stat. Phys. 80, 307–336 (1995).
    [CrossRef]
  19. V. Balakrishnan, C. Nicolis, and G. Nicolis, “Recurrence time statistics in deterministic and stochastic dynamical systems in continuous time: A comparison,” Phys. Rev. E 61, 2490–2499 (2000).
    [CrossRef]
  20. H. Solari, E. Eschenazi, R. Gilmore, and J. R. Tredicce, “Influence of coexisting attractors on the dynamics of a laser system,” Opt. Commun. 64, 49–53 (1987).
    [CrossRef]
  21. I. B. Schwartz and T. W. Carr, “Bi-instability as a precursor to global mixed-mode chaos,” Phys. Rev. E 59, 6658–6661 (1999).
    [CrossRef]
  22. F. T. Arecchi, R. Meucci, G. Puccioni, and J. R. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switch gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
    [CrossRef]
  23. T. Midavaine, D. Dangoisse, and P. Glorieux, “Observation of chaos in a frequency modulated CO2 laser,” Phys. Rev. Lett. 55, 1989–1992 (1985).
    [CrossRef] [PubMed]
  24. I. I. Matorin, A. S. Pikovskii, and Ya. I. Khanin, “Multistability and autostochasticity in a laser with a delayed-response active medium subjected to periodic loss modulation,” Sov. J. Quantum Electron. 14, 1401–1405 (1984)
    [CrossRef]
  25. J. R. Tredicce, F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, “Dynamic behavior and onset of low dimensional chaos in a modulated homogeneosly broadened single mode laser: Experiment and theory,” Phys. Rev. A 34, 2073 (1986).
    [CrossRef] [PubMed]
  26. G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, “Measurement of the formation and evolution of a strange attractor in a laser,” Phys. Rev. Lett. 55, 339–342 (1985).
    [CrossRef] [PubMed]
  27. D. Hennequin, P. Glorieux, and D. Dangoisse, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1987).
  28. R. Meucci, A. Poggi, F. T. Arecchi, and J. R. Tredicce, “Dissipativity of an optical chaotic system characterized via generalized multistability,” Opt. Commun. 65, 151–156 (1988).
    [CrossRef]
  29. R. Gilmore and M. Lefranc, The Topology of Chaos. Alice in Stretch and Squeezeland (Wiley Interscience, 2002), p. 8.
  30. H. L. D. S. Cavalcante, M. Oria, D. Sornette, E. Ott, and D. J. Gauthier, “Predictability and suppression of extreme events in a chaotic system,” Phys. Rev. Lett. 111, 198701 (2013).
    [CrossRef] [PubMed]

2013

A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
[CrossRef]

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 87, 2339108 (2013).

J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
[CrossRef]

C. Bonazzola, A. Hnilo, M. Kovalsky, and J. R. Tredicce, “Optical rogue waves in an all-solid state laser with saturable absorber: importance of spatial effects,” J. Opt. 15, 064004 (2013).
[CrossRef]

H. L. D. S. Cavalcante, M. Oria, D. Sornette, E. Ott, and D. J. Gauthier, “Predictability and suppression of extreme events in a chaotic system,” Phys. Rev. Lett. 111, 198701 (2013).
[CrossRef] [PubMed]

2012

S. K. El-Labany, W. M. Moslem, N. A. El-Bedwehy, R. Sabry, and H. N. A. El-Razek, “Rogue waves in Titan’s atmosphere,” Astrophys. Space Sci. 338, 3–8 (2012).
[CrossRef]

H. Kawamura, T. Hatano, N. Kato, S. Biswas, and B. K. Chakrabartik, “Statistical physics of fracture, friction, and earthquakes,” Rev. Mod. Phys. 84, 839–884 (2012).
[CrossRef]

2011

C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[CrossRef] [PubMed]

M. Kovalsky, A. Hilo, and J. R. Tredicce, “Extreme events in the Ti:sapphire laser,” Opt. Lett. 36, 4449–4451 (2011)
[CrossRef] [PubMed]

2010

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

2008

S. Aberg and G. Lindgren, “Height distribution of stochastic Lagrange ocean waves,” Prob. Eng. Mech. 23, 359–363 (2008).
[CrossRef]

2007

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

2004

U. F. de Pinho, P. C. Liu, and C. E. P. Ribeiro, “Freak waves at Campos Basin, Brazil,” Geofizika 21, 53 (2004)

2000

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Recurrence time statistics in deterministic and stochastic dynamical systems in continuous time: A comparison,” Phys. Rev. E 61, 2490–2499 (2000).
[CrossRef]

1999

I. B. Schwartz and T. W. Carr, “Bi-instability as a precursor to global mixed-mode chaos,” Phys. Rev. E 59, 6658–6661 (1999).
[CrossRef]

1995

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Extreme value distributions in chaotic dynamics,” J. Stat. Phys. 80, 307–336 (1995).
[CrossRef]

1988

P. Gaspard and X.-J. Wang, “Sporadicity: Between periodic and chaotic dynamical behaviors,” Proc. Natl. Acad. Sci. USA 85, 4591–4595 (1988).
[CrossRef] [PubMed]

R. Meucci, A. Poggi, F. T. Arecchi, and J. R. Tredicce, “Dissipativity of an optical chaotic system characterized via generalized multistability,” Opt. Commun. 65, 151–156 (1988).
[CrossRef]

1987

D. Hennequin, P. Glorieux, and D. Dangoisse, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1987).

H. Solari, E. Eschenazi, R. Gilmore, and J. R. Tredicce, “Influence of coexisting attractors on the dynamics of a laser system,” Opt. Commun. 64, 49–53 (1987).
[CrossRef]

1986

J. R. Tredicce, F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, “Dynamic behavior and onset of low dimensional chaos in a modulated homogeneosly broadened single mode laser: Experiment and theory,” Phys. Rev. A 34, 2073 (1986).
[CrossRef] [PubMed]

1985

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, “Measurement of the formation and evolution of a strange attractor in a laser,” Phys. Rev. Lett. 55, 339–342 (1985).
[CrossRef] [PubMed]

T. Midavaine, D. Dangoisse, and P. Glorieux, “Observation of chaos in a frequency modulated CO2 laser,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

1984

I. I. Matorin, A. S. Pikovskii, and Ya. I. Khanin, “Multistability and autostochasticity in a laser with a delayed-response active medium subjected to periodic loss modulation,” Sov. J. Quantum Electron. 14, 1401–1405 (1984)
[CrossRef]

1982

F. T. Arecchi, R. Meucci, G. Puccioni, and J. R. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switch gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Aberg, S.

S. Aberg and G. Lindgren, “Height distribution of stochastic Lagrange ocean waves,” Prob. Eng. Mech. 23, 359–363 (2008).
[CrossRef]

Akhmediev, N.

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 87, 2339108 (2013).

Arecchi, F. T.

R. Meucci, A. Poggi, F. T. Arecchi, and J. R. Tredicce, “Dissipativity of an optical chaotic system characterized via generalized multistability,” Opt. Commun. 65, 151–156 (1988).
[CrossRef]

J. R. Tredicce, F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, “Dynamic behavior and onset of low dimensional chaos in a modulated homogeneosly broadened single mode laser: Experiment and theory,” Phys. Rev. A 34, 2073 (1986).
[CrossRef] [PubMed]

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, “Measurement of the formation and evolution of a strange attractor in a laser,” Phys. Rev. Lett. 55, 339–342 (1985).
[CrossRef] [PubMed]

F. T. Arecchi, R. Meucci, G. Puccioni, and J. R. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switch gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Balakrishnan, V.

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Recurrence time statistics in deterministic and stochastic dynamical systems in continuous time: A comparison,” Phys. Rev. E 61, 2490–2499 (2000).
[CrossRef]

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Extreme value distributions in chaotic dynamics,” J. Stat. Phys. 80, 307–336 (1995).
[CrossRef]

Barland, S.

J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
[CrossRef]

C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[CrossRef] [PubMed]

Biswas, S.

H. Kawamura, T. Hatano, N. Kato, S. Biswas, and B. K. Chakrabartik, “Statistical physics of fracture, friction, and earthquakes,” Rev. Mod. Phys. 84, 839–884 (2012).
[CrossRef]

Bonatto, C.

C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[CrossRef] [PubMed]

Bonazzola, C.

C. Bonazzola, A. Hnilo, M. Kovalsky, and J. R. Tredicce, “Optical rogue waves in an all-solid state laser with saturable absorber: importance of spatial effects,” J. Opt. 15, 064004 (2013).
[CrossRef]

Borges, J. L.

J. L. Borges, “Siete Noches” - La Divina comedia; Nueve Ensayos Dantescos, Ed. Fondo de Cultura Economica, Buenos Aires (1980).

Carr, T. W.

I. B. Schwartz and T. W. Carr, “Bi-instability as a precursor to global mixed-mode chaos,” Phys. Rev. E 59, 6658–6661 (1999).
[CrossRef]

Cavalcante, H. L. D. S.

H. L. D. S. Cavalcante, M. Oria, D. Sornette, E. Ott, and D. J. Gauthier, “Predictability and suppression of extreme events in a chaotic system,” Phys. Rev. Lett. 111, 198701 (2013).
[CrossRef] [PubMed]

Chakrabartik, B. K.

H. Kawamura, T. Hatano, N. Kato, S. Biswas, and B. K. Chakrabartik, “Statistical physics of fracture, friction, and earthquakes,” Rev. Mod. Phys. 84, 839–884 (2012).
[CrossRef]

Collins, K.

A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
[CrossRef]

Dangoisse, D.

D. Hennequin, P. Glorieux, and D. Dangoisse, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1987).

T. Midavaine, D. Dangoisse, and P. Glorieux, “Observation of chaos in a frequency modulated CO2 laser,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

de Pinho, U. F.

U. F. de Pinho, P. C. Liu, and C. E. P. Ribeiro, “Freak waves at Campos Basin, Brazil,” Geofizika 21, 53 (2004)

El-Bedwehy, N. A.

S. K. El-Labany, W. M. Moslem, N. A. El-Bedwehy, R. Sabry, and H. N. A. El-Razek, “Rogue waves in Titan’s atmosphere,” Astrophys. Space Sci. 338, 3–8 (2012).
[CrossRef]

El-Labany, S. K.

S. K. El-Labany, W. M. Moslem, N. A. El-Bedwehy, R. Sabry, and H. N. A. El-Razek, “Rogue waves in Titan’s atmosphere,” Astrophys. Space Sci. 338, 3–8 (2012).
[CrossRef]

El-Razek, H. N. A.

S. K. El-Labany, W. M. Moslem, N. A. El-Bedwehy, R. Sabry, and H. N. A. El-Razek, “Rogue waves in Titan’s atmosphere,” Astrophys. Space Sci. 338, 3–8 (2012).
[CrossRef]

Eschenazi, E.

H. Solari, E. Eschenazi, R. Gilmore, and J. R. Tredicce, “Influence of coexisting attractors on the dynamics of a laser system,” Opt. Commun. 64, 49–53 (1987).
[CrossRef]

Feyereisen, M.

C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[CrossRef] [PubMed]

Finot, C.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

Gadomski, W.

J. R. Tredicce, F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, “Dynamic behavior and onset of low dimensional chaos in a modulated homogeneosly broadened single mode laser: Experiment and theory,” Phys. Rev. A 34, 2073 (1986).
[CrossRef] [PubMed]

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, “Measurement of the formation and evolution of a strange attractor in a laser,” Phys. Rev. Lett. 55, 339–342 (1985).
[CrossRef] [PubMed]

Garbin, B.

J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
[CrossRef]

Gaspard, P.

P. Gaspard and X.-J. Wang, “Sporadicity: Between periodic and chaotic dynamical behaviors,” Proc. Natl. Acad. Sci. USA 85, 4591–4595 (1988).
[CrossRef] [PubMed]

Gauthier, D. J.

H. L. D. S. Cavalcante, M. Oria, D. Sornette, E. Ott, and D. J. Gauthier, “Predictability and suppression of extreme events in a chaotic system,” Phys. Rev. Lett. 111, 198701 (2013).
[CrossRef] [PubMed]

Gilmore, R.

H. Solari, E. Eschenazi, R. Gilmore, and J. R. Tredicce, “Influence of coexisting attractors on the dynamics of a laser system,” Opt. Commun. 64, 49–53 (1987).
[CrossRef]

R. Gilmore and M. Lefranc, The Topology of Chaos. Alice in Stretch and Squeezeland (Wiley Interscience, 2002), p. 8.

Giudici, M

C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[CrossRef] [PubMed]

Giudici, M.

J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
[CrossRef]

Glorieux, P.

D. Hennequin, P. Glorieux, and D. Dangoisse, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1987).

T. Midavaine, D. Dangoisse, and P. Glorieux, “Observation of chaos in a frequency modulated CO2 laser,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

Grelu, P.

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 87, 2339108 (2013).

Hammani, K.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

Hatano, T.

H. Kawamura, T. Hatano, N. Kato, S. Biswas, and B. K. Chakrabartik, “Statistical physics of fracture, friction, and earthquakes,” Rev. Mod. Phys. 84, 839–884 (2012).
[CrossRef]

Hennequin, D.

D. Hennequin, P. Glorieux, and D. Dangoisse, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1987).

Hilo, A.

Hnilo, A.

C. Bonazzola, A. Hnilo, M. Kovalsky, and J. R. Tredicce, “Optical rogue waves in an all-solid state laser with saturable absorber: importance of spatial effects,” J. Opt. 15, 064004 (2013).
[CrossRef]

Houtani, H.

A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
[CrossRef]

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

Kato, N.

H. Kawamura, T. Hatano, N. Kato, S. Biswas, and B. K. Chakrabartik, “Statistical physics of fracture, friction, and earthquakes,” Rev. Mod. Phys. 84, 839–884 (2012).
[CrossRef]

Kawamura, H.

H. Kawamura, T. Hatano, N. Kato, S. Biswas, and B. K. Chakrabartik, “Statistical physics of fracture, friction, and earthquakes,” Rev. Mod. Phys. 84, 839–884 (2012).
[CrossRef]

Khanin, Ya. I.

I. I. Matorin, A. S. Pikovskii, and Ya. I. Khanin, “Multistability and autostochasticity in a laser with a delayed-response active medium subjected to periodic loss modulation,” Sov. J. Quantum Electron. 14, 1401–1405 (1984)
[CrossRef]

Kibler, B.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

Kinoshita, T.

A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
[CrossRef]

Kjeldsen, P.

P. Kjeldsen, “A sudden disaster in extreme waves,” Rogue Waves 2000, IFREMER (1935).

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

Kovalsky, M.

C. Bonazzola, A. Hnilo, M. Kovalsky, and J. R. Tredicce, “Optical rogue waves in an all-solid state laser with saturable absorber: importance of spatial effects,” J. Opt. 15, 064004 (2013).
[CrossRef]

M. Kovalsky, A. Hilo, and J. R. Tredicce, “Extreme events in the Ti:sapphire laser,” Opt. Lett. 36, 4449–4451 (2011)
[CrossRef] [PubMed]

Laplace, P. S.

P. S. Laplace, A Philosophical Essay on Probability, translated (John Wiley, 1902).

Lecaplain, C.

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 87, 2339108 (2013).

Lefranc, M.

R. Gilmore and M. Lefranc, The Topology of Chaos. Alice in Stretch and Squeezeland (Wiley Interscience, 2002), p. 8.

Lindgren, G.

S. Aberg and G. Lindgren, “Height distribution of stochastic Lagrange ocean waves,” Prob. Eng. Mech. 23, 359–363 (2008).
[CrossRef]

Liu, P. C.

U. F. de Pinho, P. C. Liu, and C. E. P. Ribeiro, “Freak waves at Campos Basin, Brazil,” Geofizika 21, 53 (2004)

Masoller, C.

J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
[CrossRef]

C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[CrossRef] [PubMed]

Matorin, I. I.

I. I. Matorin, A. S. Pikovskii, and Ya. I. Khanin, “Multistability and autostochasticity in a laser with a delayed-response active medium subjected to periodic loss modulation,” Sov. J. Quantum Electron. 14, 1401–1405 (1984)
[CrossRef]

Meucci, R.

R. Meucci, A. Poggi, F. T. Arecchi, and J. R. Tredicce, “Dissipativity of an optical chaotic system characterized via generalized multistability,” Opt. Commun. 65, 151–156 (1988).
[CrossRef]

F. T. Arecchi, R. Meucci, G. Puccioni, and J. R. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switch gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Midavaine, T.

T. Midavaine, D. Dangoisse, and P. Glorieux, “Observation of chaos in a frequency modulated CO2 laser,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

Moslem, W. M.

S. K. El-Labany, W. M. Moslem, N. A. El-Bedwehy, R. Sabry, and H. N. A. El-Razek, “Rogue waves in Titan’s atmosphere,” Astrophys. Space Sci. 338, 3–8 (2012).
[CrossRef]

Munk, W. H.

W. H. Munk, “Proposed uniform procedure for observing waves and interpreting instruments records,” Wave Project at the Scripps Institute of Oceanography, Ca. USA (1944).

Nicolis, C.

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Recurrence time statistics in deterministic and stochastic dynamical systems in continuous time: A comparison,” Phys. Rev. E 61, 2490–2499 (2000).
[CrossRef]

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Extreme value distributions in chaotic dynamics,” J. Stat. Phys. 80, 307–336 (1995).
[CrossRef]

Nicolis, G.

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Recurrence time statistics in deterministic and stochastic dynamical systems in continuous time: A comparison,” Phys. Rev. E 61, 2490–2499 (2000).
[CrossRef]

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Extreme value distributions in chaotic dynamics,” J. Stat. Phys. 80, 307–336 (1995).
[CrossRef]

Onorato, M.

A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
[CrossRef]

Oria, M.

H. L. D. S. Cavalcante, M. Oria, D. Sornette, E. Ott, and D. J. Gauthier, “Predictability and suppression of extreme events in a chaotic system,” Phys. Rev. Lett. 111, 198701 (2013).
[CrossRef] [PubMed]

Ott, E.

H. L. D. S. Cavalcante, M. Oria, D. Sornette, E. Ott, and D. J. Gauthier, “Predictability and suppression of extreme events in a chaotic system,” Phys. Rev. Lett. 111, 198701 (2013).
[CrossRef] [PubMed]

Picozzi, A.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

Pikovskii, A. S.

I. I. Matorin, A. S. Pikovskii, and Ya. I. Khanin, “Multistability and autostochasticity in a laser with a delayed-response active medium subjected to periodic loss modulation,” Sov. J. Quantum Electron. 14, 1401–1405 (1984)
[CrossRef]

Poggi, A.

R. Meucci, A. Poggi, F. T. Arecchi, and J. R. Tredicce, “Dissipativity of an optical chaotic system characterized via generalized multistability,” Opt. Commun. 65, 151–156 (1988).
[CrossRef]

J. R. Tredicce, F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, “Dynamic behavior and onset of low dimensional chaos in a modulated homogeneosly broadened single mode laser: Experiment and theory,” Phys. Rev. A 34, 2073 (1986).
[CrossRef] [PubMed]

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, “Measurement of the formation and evolution of a strange attractor in a laser,” Phys. Rev. Lett. 55, 339–342 (1985).
[CrossRef] [PubMed]

Proment, D.

A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
[CrossRef]

Puccioni, G.

F. T. Arecchi, R. Meucci, G. Puccioni, and J. R. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switch gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Puccioni, G. P.

J. R. Tredicce, F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, “Dynamic behavior and onset of low dimensional chaos in a modulated homogeneosly broadened single mode laser: Experiment and theory,” Phys. Rev. A 34, 2073 (1986).
[CrossRef] [PubMed]

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, “Measurement of the formation and evolution of a strange attractor in a laser,” Phys. Rev. Lett. 55, 339–342 (1985).
[CrossRef] [PubMed]

Ribeiro, C. E. P.

U. F. de Pinho, P. C. Liu, and C. E. P. Ribeiro, “Freak waves at Campos Basin, Brazil,” Geofizika 21, 53 (2004)

Rios Leite, J. R.

J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
[CrossRef]

C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[CrossRef] [PubMed]

Ropers, C.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

Sabry, R.

S. K. El-Labany, W. M. Moslem, N. A. El-Bedwehy, R. Sabry, and H. N. A. El-Razek, “Rogue waves in Titan’s atmosphere,” Astrophys. Space Sci. 338, 3–8 (2012).
[CrossRef]

Schwartz, I. B.

I. B. Schwartz and T. W. Carr, “Bi-instability as a precursor to global mixed-mode chaos,” Phys. Rev. E 59, 6658–6661 (1999).
[CrossRef]

Solari, H.

H. Solari, E. Eschenazi, R. Gilmore, and J. R. Tredicce, “Influence of coexisting attractors on the dynamics of a laser system,” Opt. Commun. 64, 49–53 (1987).
[CrossRef]

Solli, D. R.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

Sornette, D.

H. L. D. S. Cavalcante, M. Oria, D. Sornette, E. Ott, and D. J. Gauthier, “Predictability and suppression of extreme events in a chaotic system,” Phys. Rev. Lett. 111, 198701 (2013).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 87, 2339108 (2013).

Toffoli, A.

A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
[CrossRef]

Tredicce, J. R.

J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
[CrossRef]

C. Bonazzola, A. Hnilo, M. Kovalsky, and J. R. Tredicce, “Optical rogue waves in an all-solid state laser with saturable absorber: importance of spatial effects,” J. Opt. 15, 064004 (2013).
[CrossRef]

C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[CrossRef] [PubMed]

M. Kovalsky, A. Hilo, and J. R. Tredicce, “Extreme events in the Ti:sapphire laser,” Opt. Lett. 36, 4449–4451 (2011)
[CrossRef] [PubMed]

R. Meucci, A. Poggi, F. T. Arecchi, and J. R. Tredicce, “Dissipativity of an optical chaotic system characterized via generalized multistability,” Opt. Commun. 65, 151–156 (1988).
[CrossRef]

H. Solari, E. Eschenazi, R. Gilmore, and J. R. Tredicce, “Influence of coexisting attractors on the dynamics of a laser system,” Opt. Commun. 64, 49–53 (1987).
[CrossRef]

J. R. Tredicce, F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, “Dynamic behavior and onset of low dimensional chaos in a modulated homogeneosly broadened single mode laser: Experiment and theory,” Phys. Rev. A 34, 2073 (1986).
[CrossRef] [PubMed]

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, “Measurement of the formation and evolution of a strange attractor in a laser,” Phys. Rev. Lett. 55, 339–342 (1985).
[CrossRef] [PubMed]

F. T. Arecchi, R. Meucci, G. Puccioni, and J. R. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switch gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Wang, X.-J.

P. Gaspard and X.-J. Wang, “Sporadicity: Between periodic and chaotic dynamical behaviors,” Proc. Natl. Acad. Sci. USA 85, 4591–4595 (1988).
[CrossRef] [PubMed]

Waseda, T.

A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
[CrossRef]

Zamora-Munt, J.

J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
[CrossRef]

Astrophys. Space Sci.

S. K. El-Labany, W. M. Moslem, N. A. El-Bedwehy, R. Sabry, and H. N. A. El-Razek, “Rogue waves in Titan’s atmosphere,” Astrophys. Space Sci. 338, 3–8 (2012).
[CrossRef]

Geofizika

U. F. de Pinho, P. C. Liu, and C. E. P. Ribeiro, “Freak waves at Campos Basin, Brazil,” Geofizika 21, 53 (2004)

J. Opt.

C. Bonazzola, A. Hnilo, M. Kovalsky, and J. R. Tredicce, “Optical rogue waves in an all-solid state laser with saturable absorber: importance of spatial effects,” J. Opt. 15, 064004 (2013).
[CrossRef]

J. Stat. Phys.

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Extreme value distributions in chaotic dynamics,” J. Stat. Phys. 80, 307–336 (1995).
[CrossRef]

Nature

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

Opt. Commun.

R. Meucci, A. Poggi, F. T. Arecchi, and J. R. Tredicce, “Dissipativity of an optical chaotic system characterized via generalized multistability,” Opt. Commun. 65, 151–156 (1988).
[CrossRef]

H. Solari, E. Eschenazi, R. Gilmore, and J. R. Tredicce, “Influence of coexisting attractors on the dynamics of a laser system,” Opt. Commun. 64, 49–53 (1987).
[CrossRef]

Opt. Lett.

Phys. Lett. A

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

Phys. Rev. A

J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. Rios Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87, 035802 (2013).
[CrossRef]

J. R. Tredicce, F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, “Dynamic behavior and onset of low dimensional chaos in a modulated homogeneosly broadened single mode laser: Experiment and theory,” Phys. Rev. A 34, 2073 (1986).
[CrossRef] [PubMed]

Phys. Rev. E

V. Balakrishnan, C. Nicolis, and G. Nicolis, “Recurrence time statistics in deterministic and stochastic dynamical systems in continuous time: A comparison,” Phys. Rev. E 61, 2490–2499 (2000).
[CrossRef]

I. B. Schwartz and T. W. Carr, “Bi-instability as a precursor to global mixed-mode chaos,” Phys. Rev. E 59, 6658–6661 (1999).
[CrossRef]

A. Toffoli, T. Waseda, H. Houtani, T. Kinoshita, K. Collins, D. Proment, and M. Onorato, “Excitation of rogue waves in a variable medium: An experimental study on the interaction of water waves and currents,” Phys. Rev. E 87051201 (2013).
[CrossRef]

Phys. Rev. Lett.

C. Bonatto, M. Feyereisen, S. Barland, M Giudici, C. Masoller, J. R. Rios Leite, and J. R. Tredicce, “Deterministic optical rogue waves,” Phys. Rev. Lett. 107, 053901 (2011).
[CrossRef] [PubMed]

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 87, 2339108 (2013).

F. T. Arecchi, R. Meucci, G. Puccioni, and J. R. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability and turbulence in a Q-switch gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

T. Midavaine, D. Dangoisse, and P. Glorieux, “Observation of chaos in a frequency modulated CO2 laser,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, “Measurement of the formation and evolution of a strange attractor in a laser,” Phys. Rev. Lett. 55, 339–342 (1985).
[CrossRef] [PubMed]

D. Hennequin, P. Glorieux, and D. Dangoisse, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1987).

H. L. D. S. Cavalcante, M. Oria, D. Sornette, E. Ott, and D. J. Gauthier, “Predictability and suppression of extreme events in a chaotic system,” Phys. Rev. Lett. 111, 198701 (2013).
[CrossRef] [PubMed]

Prob. Eng. Mech.

S. Aberg and G. Lindgren, “Height distribution of stochastic Lagrange ocean waves,” Prob. Eng. Mech. 23, 359–363 (2008).
[CrossRef]

Proc. Natl. Acad. Sci. USA

P. Gaspard and X.-J. Wang, “Sporadicity: Between periodic and chaotic dynamical behaviors,” Proc. Natl. Acad. Sci. USA 85, 4591–4595 (1988).
[CrossRef] [PubMed]

Rev. Mod. Phys.

H. Kawamura, T. Hatano, N. Kato, S. Biswas, and B. K. Chakrabartik, “Statistical physics of fracture, friction, and earthquakes,” Rev. Mod. Phys. 84, 839–884 (2012).
[CrossRef]

Sov. J. Quantum Electron.

I. I. Matorin, A. S. Pikovskii, and Ya. I. Khanin, “Multistability and autostochasticity in a laser with a delayed-response active medium subjected to periodic loss modulation,” Sov. J. Quantum Electron. 14, 1401–1405 (1984)
[CrossRef]

Other

W. H. Munk, “Proposed uniform procedure for observing waves and interpreting instruments records,” Wave Project at the Scripps Institute of Oceanography, Ca. USA (1944).

P. Kjeldsen, “A sudden disaster in extreme waves,” Rogue Waves 2000, IFREMER (1935).

J. L. Borges, “Siete Noches” - La Divina comedia; Nueve Ensayos Dantescos, Ed. Fondo de Cultura Economica, Buenos Aires (1980).

P. S. Laplace, A Philosophical Essay on Probability, translated (John Wiley, 1902).

R. Gilmore and M. Lefranc, The Topology of Chaos. Alice in Stretch and Squeezeland (Wiley Interscience, 2002), p. 8.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

(a) Bifurcation diagram graphing the maxima of intensity S as a function of the modulation amplitude m. The parameters used in the numerical integration are: A= 1.30, β = 0.09441, and γ = 0.027. (b) Phase space portrait, Intensity S as a function of population inversion D for m = 0.68. All other parameters values are the same as in Fig. 1(a).

Fig. 2
Fig. 2

Histogram of the calculated maxima of intensity corresponding to the time signal of Fig. 1(b). The vertical dashed line indicates the threshold corresponding to the AI calculated as the mean value plus four times the standard deviation of the probability distribution. Clearly no “optical rogue events” occur.

Fig. 3
Fig. 3

Blow-up of lower portion of the bifurcation diagram of Fig. 1(a). This figure shows how the density of pulse maxima increases, accumulating near zero value.

Fig. 4
Fig. 4

Bifurcation diagram as in Fig. 1(a) but with the following parameter values: A=1.3; β = 0.0135; and γ = 0.00027.

Fig. 5
Fig. 5

Long series of pulse Intensity S as a function of time measured in units of the modulation period. The parameter values are the same as in Fig. 3 with m = 0.01236. This time series contains rare abnormally large amplitude spikes.

Fig. 6
Fig. 6

Pulse Intensity S as a function of time measured in units of the modulation period. This is a time zoom of Fig. 5 (with displaced time scale).

Fig. 7
Fig. 7

(a) Probability distribution of the maxima intensity of the pulses for a modulation amplitude m = 0.01215; all other parameter values are the same as in Fig. 4, (b) same as Fig. 7(a) for a modulation amplitude m=0.01225.

Fig. 8
Fig. 8

Number of events (RP) overcoming the AI as a function of the modulation amplitude m. Before the crisis at m = 0.0122 extreme events have not been observed. After the crisis the number of extreme events grows, reaches a maximum, and decreases back to zero at m = 0.0125.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

d I ( t ) d t = k ( t ) I ( t ) + g I ( t ) N ( t )
d N ( t ) d t = γ ( N ( t ) N 0 ) 2 g I ( t ) N ( t )
k ( t ) = k 0 ( 1 + m cos ( ω t ) )
Ω = [ k 0 γ ( A 1 ) γ 2 A 2 / 4 ] 1 / 2
d S / d τ = S [ 1 + m cos ( β τ ) D ]
d D / d τ = γ [ D A + S D ]

Metrics